Each object can be assigned to a point group according to its symmetry elements:

Transcription

1 3. Symmetry and Group Theory READING: Chapter Point groups Each object can be assigned to a point group according to its symmetry elements: Source: Shriver & Atkins, Inorganic Chemistry, 3 rd Edition. The symmetry elements can be considered operators. Operator Instruction for an operation to be performed on a function or object that follows it: 64

2 3.. Group theory in a nut-shell Group Theory is the mathematical treatment of the properties of groups (e.g., point groups!). Powerful mathematical tool used to simplify quantum mechanical calculations using molecular symmetry. Ultimately, it can help generate molecular orbitals and predict the spectroscopic characteristics of a molecule using back-of-an-envelope calculations. There are entire textbooks devoted to chemical applications of Group Theory, but we will just look at a brief introduction to the subject the rest you can learn in a spectroscopy course or in grad school! Group theory is used to construct character tables, e.g. C V (See Resource Section 4 of your Textbook for a complete set of character tables.) A quick walk around a character table The symbol in the top left shows the point group to which the character table applies (C v in this case). The symbols across the top show the symmetry operations. A molecule with C v symmetry (e.g., H O) will have a C axis of rotation and two mirror planes (σ v and σ v ), but it will not have a center of inversion (no i operation shows up in this table). The symbols down the left side (A, A, B, B ) are Mulliken symbols, denoting irreducible representations (we ll get to that ) The symbol in the bottom left corner (Γ x,y,z ) is a reducible representation (we ll get to that, too ) The numbers in the table provide specific information about the relationship between the representations and the symmetry operations (we ll see this shortly). 65

3 So let s have a look at how symmetry operations can be treated as groups Consider the v point group: e.g., NH 3 v has six different symmetry operators {E,,, σ v, σ v, σ v } These operators can be combined into products, e.g.: x = σ v x σ v = For the set of operators we can define a multiplication table. The multiplication table describes all products of any two operators in our sample set: E σ v σ v σ v E E σ v σ v σ v E σ v σ v σ v E σ v σ v σ v σ v σ v σ v σ v E σ v σ v σ v σ v E σ v σ v σ v σ v E The above operator multiplication table fulfills the four criteria that define a group and so do the symmetry operators that define ANY point group! The four criteria that define a group are: ) In each group there exists an operator that commutes with all other operators in the group and leaves them unchanged: E x = x E = E is the identity operator (neutral element) of the group (from the German Einheitsoperator = unity operator) ) The product of two group elements must also be an element of the group. No new operators mysteriously appear! i.e. the group is closed 66

4 x σ v = σ v σ v x σ v = E etc. (see table) 3) The products are associative, i.e. (X x Y) x Z = X x (Y x Z) for all elements x (σ v x σ v ) = ( x σ v ) x σ v x = σ v x σ v E = E 4) For each element Z a reciprocal element Z - exists. An element and its reciprocal commute. i.e., Z x Z - = Z - Z = E There must be a reciprocal element for such that: x [ ] - = E See multiplication table to find: x = E Therefore is the reciprocal element of : = [ ] - Question: Which is the reciprocal element of σ v? Def.: A group in which the order of multiplication of the elements is irrelevant is called an Abelian group. Cyclic point groups are typically Abelian, others are usually not. The v point group is not Abelian, because x σ v σ v x Def.: If a subset of a group is a group by itself, it is called a subgroup. e.g., The following table is a subgroup of our example multiplication table: E E E E E Def.: The number of elements in a (sub)group is called it order. - The order of the v point group is 6. - The order of the above subgroup is 3. Def.: Two group elements X and Y are conjugate, if the following equality holds for any X and Z: 67

5 Z - x X x Z = Y We say that X and Y are conjugate. We say that Y is the similarity transform of X by Z. This equality is called a similarity transformation. [ ] - x σ v x x σ v x σ v x = σ v So, σ v and σ v are conjugate and σ v is the similarity transform of σ v by Def.: A class is a complete set of operators that are conjugated to each other e.g. {E},{, } and { σ v, σ v, σ v } in the v point group. The number of elements in a class is an integral divider of the group order. HOMEWORK: Prove to yourself that ALL POSSIBLE similarity transforms of σ v are members of the { σ v, σ v, σ v } class. HOMEWORK: For boric acid (shown below), determine all applicable symmetry operations and create a multiplication table. Prove that these symmetry operations meet all four criteria to form a group. To which point group does boric acid belong? Is this an Abelian point group? Boric acid 68

6 3.3. Matrix Representations of Symmetry Operations Mathematically, the best way to understand how the symmetry operations of a molecule influence its properties is to study the sets of matrices which mirror, by their group table, those same operations. Such a set of matrices are said to form a representation of the point group. Essentially, when we introduce a matrix representation, we are replacing the geometry of symmetry operations with the algebra of matrices. 69

7 Brief review of matrix algebra Matrix: Rectangular array of numbers here we will only need quadratic matrices. i.e., # rows = # columns: Multiplication of matrix with a vector x (Dot products!) y m = a mj x j (row x column) j A x = Y i.e., y = a x + a x + + a n x n e.g.: Product of two matrices A X = Y i.e., y = a x + a x + a 3 x a n x n e.g.: 70

8 Any symmetry operation σ, i, C n, etc. can mathematically be represented by a matrix. Consider point group C h, which consists of the operations E, σ xy, i, C What do these operations do to a general vector (x, y, z)? Let s operate on the vector r = (x, y, z) Identity operator E: Applying the identity operator E to a vector (x, y,z) should give us back the vector (x,y,z). The same result is achieved by multiply the vector by the above 3x3 matrix. CONCLUSION: The identity operator E can be mathematically represented by the above 3x3 matrix! Inversion center i: Mirror plane σ xy : 7

9 Rotation C : a bit more complicated. Let s derive the matrix for a general rotation of vector r of length l into vector r first: Recall your high school trigonometry: sin α = opposite/hypotenuse cos α = adjacent/hypotenuse x = l cos α y = l sin α x = l cos (α + θ) y = l sin (α + θ) Using the trigonometric addition theorems: cos (α + θ) = cos θ cos α - sin θ sin α sin (α + θ) = sin θ cos α + cos θ sin α x = l cos θ cos α - l sin θ sin α = x cos θ - y sin θ y = l sin θ cos α + l cos θ sin α = x sin θ + y cos θ Written as a matrix: For C θ = 80 cos θ = - and sin θ = 0 In all three dimensions the matrix representing C therefore is: 7

10 The matrices that represent the symmetry operations that define C h form a group. The matrices and symmetry operators are isomorphous (have the same form). Multiplication table for C h : E C σ h i E E C σ h i C C E i σ h σ h σ h i E C i i σ h C E e.g.: C x σ h = i: Some more general properties of matrices we need to know: The product of two diagonal matrices (all elements a ij with i j = 0) is also a diagonal matrix, i.e.: A block diagonal matrix consists of submatrices of quadratic shape; the product of two block diagonalized matrices is again a block diagonalized matrix of the same form: 73

11 Def.: For any cooperative matrix there exists an inverse matrix such that Q x Q - = where = unit matrix inverse elements in multiplication tables!!! Using inverse matrices we can perform similarity transformations with matrices: B = Q - x A x Q We say matrices A and B are conjugated. Reducible and irreducible representations Consider a set of matrices {A, B, C} that represent the symmetry operations of a point group. Similarity transformations can be performed on them until we arrive at a set of block diagonalized matrices {A, B, C }, i.e.: A = Q - x A x Q B = Q - x B x Q C = Q - x C x Q These matrices {A, B, C } are still representations of the same operations represented by {A, B, C}, e.g. if A x B = C then A x B = C must also be true. Proof (direct): A x B = (Q - x A x Q) x (Q - x B x Q) = Q - x A x (Q x Q - ) x B x Q = Q - x A x B x Q = Q - x C x Q = C 74

12 The block-diagonalized matrices look like this which means A x B = C ; A x B = C ; A 3 x B 3 = C 3 which means that the individual blocks of the block diagonalized matrix behaves just like the entire matrix itself!!! Each small block A n, B n, C n of the matrices A, B, C is a complete new reduced representation of the same operation as represented by the entire matrices A, B, C (or A, B, C) the overall information content is the same! A n, B n, C n fulfill the same multiplication table as the A, B, C (or A, B, C), i.e., they also are a valid representation of the group. If the representations A n, B n, C n cannot be further simplified by similarity transformations they are called irreducible representations of the particular symmetry operation to which they relate. For each point group there exists an infinite number of reducible representations, but only one finite set of irreducible representations. The irreducible representation are denoted by Mulliken symbols: Symbol A B E T Property symmetric under n-fold rotation anti-symmetric under n-fold rotation two-dimensional three-dimensional Index Position Property below symmetric under σ v or C C n below anti-symmetric under σ v or C C n g below symmetric under i u below anti-symmetric under i above symmetric under σ h if i is not present above anti-symmetric under σ h if i is not present + above symmetric under σ v in D h - above anti-symmetric under σ v in D h 75